Detecting zero-point fluctuations with stochastic Brownian oscillators

Abstract

High-quality quantum oscillators are preferred for precision sensing of external physical parameter because if the noise level due to interactions with the environment is too high, metrological information can be lost due to quantum decoherence. On the other hand, stronger interactions with a thermal environment could be seen a resource for new types of metrological schemes. We present a general amplification strategy that enables the detection zero-point fluctuations using low-quality quantum oscillators at finite temperature. We show that by injecting a controllable level of multiplicative frequency noise in a Brownian oscillator, quantum deviations from the virial theorem can be amplified by a parameter proportional to the strength of the frequency noise at constant temperature. As an application, we suggest a scheme in which the virial ratio is used as a witness of the quantum fluctuations of an unknown thermal bath, either by measuring the oscillator energy or the heat current flowing into an ancilla bath. Our work expands the metrological capacity of low-quality oscillators and can enable new measurements of the quantum properties of thermal environments by sensing their zero-point contributions to system variables.

Figures (3)

• Figure 1: Scheme of the scenario. A nonequilibrium Brownian oscillator of mass $M$ and frequency $\Omega$ is subjected to white frequency noise (given by the stochastic variable $\varphi$ and characterized by the strength $D$). The oscillator is coupled to two Ohmic baths characterized by damping constants $\Gamma_{1}$ and $\Gamma_{2}$, stochastic force $\xi_{1,2}$ and large cutoff frequency ($\Omega,\Gamma_{1,2}\ll\Omega_{\rm C}$). Ancilla bath is assumed to introduce strong dissipation ($\Gamma_{1}\lesssim\Omega$). The nature of the target bath is assumed unknown.
• Figure 2: Virial ratio $\mathcal{R}=\langle K\rangle/\langle V\rangle$ as a function of the normalized temperature $\widetilde{T}$ for an oscillator with quality factor $Q=10,20,40$ (solid, long dashed, short dashed) and a cutoff frequency for the bath $Q_{\rm C}=10^{3}$. The blue curves correspond to the full expression for a Brownian oscillator in the steady state under the influence of a quantum thermal bath, including zero-point fluctuations. Red curves correspond to the lower bound on $R$ obtained from applying the Heisenberg uncertainty principle (shaded area). The green solid line corresponds to the value of the ratio for a Brownian oscillator under the influence of a 'classical' thermal bath.
• Figure 3: (a) Net energy deviation $\Delta=\mathcal{W}(1-\mathcal{F})$ over the stability region of an oscillator at temperature $\widetilde{T}=1/4$ as a function of $D\Omega$ and $Q$, with $u_{\rm C}=10^{3}$. (b) Difference of the virial factor $\Delta\mathcal{F}_{1}=|\mathcal{F}_{\rm Q}-\mathcal{F}_{\rm cl}|$ as a function of the normalized temperature $\widetilde{T}$ for the single bath case when the bath is either quantum of classical, with $Q=10$ and $u_{\rm C}=10^{3}$. (c) Variation of the virial factor $\Delta\mathcal{F}_{2}=|\mathcal{F}_{2,\rm Q}-\mathcal{F}_{2,\rm cl}|$ for the two baths scenario with temperatures $T_{1,2}$ and different damping rates such that the relative damping is defined $\gamma\equiv\Gamma_{1}/(\Gamma_{1}+\Gamma_{2})$. The variation is considered between scenarios where the ancilla bath is fixed to be quantum and the target is either quantum or classical. The dashed vertical line corresponds to thermal equilibrium ($T_{1}=T_{2}$), with $\widetilde{T}_{1}=1/4$, $Q=10$ and $u_{\rm C}=10^{3}$.